Category Archives: Number Sense

Most students multiplied 5s, but did not talk about exponentiation — instead it was 25 groups of 5, or 5 groups of 25, or other patterns

A few students had some really creative ways of breaking down the pattern into 5s –try and figure out

8 + 9 + 2*4
4*4 + 1 * 4 + 4
4*4 + 2*2 + 5

I was really impressed with several students who shared strategies that were a bit slower, for instance counting every group of 5, without shame. I hope I can keep that culture going.

My students have already seen exponents for a few years (I teach 8th grade), and I’m curious how this would develop thinking differently for students who haven’t articulated the idea of an exponent. There’s definitely some element of intellectual need for it, but at the same time it doesn’t scream “this will make your life easier in the future”, which is something I want students to see when exploring a new concept.

Either way, kids really liked this, and there aren’t enough great visualizations of exponents out there. I’m excited to show them this one:

Several students doubled, and then doubled again to make multiplying by 4 easier

Using the distributive property to break apart 10 and the half seems like common sense, but it was a big hit in two classes. I’m beginning to think that a flexible understanding of the distributive property is a huge part of number sense

Among students who prefer to operate with improper fractions, there is a divide between students who see the opportunity to simplify the fraction first (21/2 * 4/1 = 21/1 * 2/1) rather than multiplying (one student converted to a common denominator of 4, for 84/4, then multiplied 84 * 4 and divided by 4*1). These are the valuable shortcuts that a) make calculation easier, but b), and more importantly, show an understanding of algebraic structure, in particular the critical importance of the commutative and associative properties when working with fractions.

On this note, I’ve been thinking more about the longitudinal structure of number talks. I’m structuring them day by day pretty randomly — whatever I think will be meaningful is what we think about. I’m curious if grouping them by structures I want students to see would be valuable. On the one hand, sustained practice with a mathematical idea could help solidify that idea, especially in lower-skilled students. On the flip side, math is math, and number talks as mixed practice have a lot of value in messaging math for math’s sake. On top of that, I’m not sure students could keep track of a weekly theme for number talks, the unit we’re in for the rest of class, and then a range of topics for warm-ups and the rest of their classes in a meaningful way. But maybe it’s something to think about.

A few tangents tonight. This great post from Number Loving Beagle got me thinking, in particular this piece:

I hear occasionally from teachers that we need to teach kids “responsibility” and we can’t force them to learn if they don’t want to. This line of thinking bothers me a great deal as places the burden of being eager to learn on the student. Some kids place “learning at school” very low on their priority list. We must acknowledge that rather than disregard it with “he/she never came and asked a question.” If we are being honest with ourselves, we know exactly which kids need the help but won’t outwardly seek it. We know which kids won’t ask questions when they have them, and which ones won’t make an effort to turn in assignments that they’ve missed. It’s not that they are incapable of seeking help, asking questions, and turning in assignments. But by stating that “help was offered but not taken” we do not absolve ourselves from the responsibility to reach these students.

I’ve been guilty of this. It perpetuates the haves/have nots divide in students — those who are invested in school, see the positive effects of their hard work, and continue to work hard for their education; and those who struggle to get invested and are continually discouraged by negative feedback and internalize that they are out of control of their learning. That’s one big issue, and one I care a lot about, as I think my biggest failures this year have been in a few students I’ve struggled to give reasons to engage in my class.

This got me thinking about number sense as well. We talk about number sense in these same terms — either students have it or they don’t, and especially in upper middle school we give little thought to teaching number sense. When we do “teach” number sense, it’s often asking students if their answer makes sense, or showing students a shortcut, or giving students worksheets of “basic skills” (decimal and fraction operations).

But students who have number sense gained it by making sense of math problems — by constantly checking their thought process and their answers against a mental image of the problem and solution space, by using multiple representations to understand a concept, by estimating quickly and strategically. Asking students who struggle with these skills to do them without support is asking them to fail and revert to their answer-getting ways that have got them this far.

So what does a lesson look like that supports all students, no matter where their number sense is? I have no idea. Number talks are all I have right now, but I’m looking.

Took a day off from my #MTBoS30 challenge yesterday as I was hiking in New Hampshire with 8 of our boys. Beautiful view from the top despite rain in the morning:

There wasn’t a ton of math happening, but one thing was surprising, and a bit worrying on the theme of number sense. The hike was 5 miles round trip. 2 miles up, including a number of sections like this:

And 3 miles back down. The first mile, both up and down, was less than half an hour for the group. But the last half-mile to the summit, mostly climbing up steep trail like the picture above, took over an hour. My students were very attentive to the signs marking mileage coming up, and weren’t too focused on how long it was taking. Heading down, however, they really struggled to get an idea of how far we were from the trailhead, or how long that would take. They didn’t think that 3 miles was a big deal as we started on the trail down, but were surprised to hear that we probably had more than an hour left after making about a mile of the way down. Then, upon hearing that they were still a mile from the bottom, several students were horrified that they were so far away, unaware of how fast we were moving on the relatively flat lower portion of the trail.

This isn’t a big deal, or a red flag for me mathematically. This type of number sense is tough for many adults, and we were tired and impatient after a long hike. That said, we spent some time this year talking about distance-speed-time questions, and while they often struggled with concepts, it’s a skill I would love them to take to high school, and I didn’t see it in practice. It makes me curious what kind of concrete activities could give students a sense of space and time that they could apply to their everyday lives.

Heading on an overnight hiking trip with a group of students after school today, so my post will be quick. Two different takes on avoidance in math class — Mathy McMatherson on answer-getting and everything that goes with this mindset in math class, and one of my favorite videos on teaching, Steven Leinwand’s Ignite talk on excellent instruction, which I thought of because he begins it by talking about how to not have math anxiety.

I’m not sure how to connect all of this, but today I’m thinking about routines and the ways they encourage or discourage number sense. There are plenty of things I do in my classroom that I think may discourage number sense, and I’m curious how students interpret and internalize them. I’m thinking in particular about feedback — what types of feedback encourage the answer-getting mindset, and what types discourage it? What types of feedback promote students making sense of mathematics? What types of feedback promote student discourse?

Wow. Quite a day. State testing (here, MCAS) is tomorrow. I believe teachers set the tone for kids, and I’m doing my best to make them feel calm and centered heading into tomorrow.

DivisionNumber talk today:
A few surprises:
How many student found the precise answer
How many of those students (almost all) expressed their answer as a decimal
How many of the students who chose to use an estimation strategy are my top students
How many students chose to think about it in terms of multiplying 9 by an unknown to get 1001, rather than division.

The last one is the most interesting to me. Division is defined (in rigorous mathematics, anyway) as no more than the inverse of multiplication). I think I’m happy that that is deep in the number sense of a number of my students. That said, division has got to be one of the basic procedures students do day in and day out that has the most possible representations and interpretations. Need to find some more number talks to get at that ambiguity. Makes me think of this number talk, from Fawn, and how it could be adapted for a visualization of division.

Axes
This question was the focus of class todayThese 8th graders are having a really tough time with linear equations on axes with a scale other than 1 to 1. They love counting boxes to find slope, and get confused when that doesn’t work. And this is our fault — spending too much time on that procedure, and too many questions out of context (like the one above!) And it’s great that they can reliably find slope on a conventional coordinate plane, but when we think about concepts that students will apply in the real world, finding the slope on an Algebra textbook-style coordinate plane will probably never happen. The application of the principle of rate of change absolutely will. Which one are we preparing kids for? And what does preparing kids for the second one look like? That’s my tough question of the day. It’s a critical form of number sense, and I’m worried my students don’t have it and I don’t know how to teach it beyond saying here, think about this.

Inspired by his talk at NCTM, I’ve been showing some of his nerdier comics to my students. I planned a bit of a shorter lesson today, and just planned to give the nerd search to a few students when they finished their problem set, but it ended up engaging a ton of students and inspiring a spirited discussion to end class. One student came up to me after class and asked how many she had to do (she’s a very conscientious student, but not usually especially motivated to work when she doesn’t need to). I told her it wasn’t homework, but she should try as many as she could if it made her happy. She said she wanted to get all of them, and made me explain the integral and sum. Then, feeling outworked by my students, I spent 20 minutes finding the square root of 375,559,383,241 by guess and check. I was successful! And grateful for the invention of calculators.

My current practice is to give students time to silently and mentally find an answer (in this case about 45 seconds), then we share out every answer before sharing strategies. In each class I got 6-10 different answers, and at least half of the hands shot up to share a strategy (I don’t know about your class, but that was pretty good for me).

A few of the strategies below for your enjoyment (I’m only including the correct ones, although the wrong ones were interesting as well, especially the fact that 10×10 – 7×7 came up several times).
4×10 + 5×5+ 10 + 5
4×10 + 10 + 6×4 + 11
5×10 + 5×4+ 5
10×10- 5×5
10×10- (1/4)*10×10
4×16 + 11
And plenty more, including one student who told me she counted every dot.

I was pleasantly surprised with the engagement in asking 8th graders to count dots on a page. Finding multiplicative structure in patterns of dots reflects the habits of mind students need to find structure in any mathematics — and to believe in their ability to find that structure.

I like it a lot, and I’m sure I’ll use it at some point. However, I value number talks because they promote flexible thinking with basic skills. I’m not sure how many different ways this can be interpreted — or, building off of that, another exponential pattern like this one (how many black triangles are there)
I’m curious what kids can do with these. Will post back when that happens.